Averages -- Colm Connaughton

Ergodicity is about two different types of averages, the time averages and the ensemble average, with very specific physical meanings. There are, however, many other averages, and depending on the physical meaning we aim to capture with an average we will choose one or another.
Colm reminds us of how many choices we make -- implicitly or explicitly -- before arriving at an average.
The paper about the mean, mentioned at 11:05min is available on the arXiv at arxiv.org/abs/...

[@guitarmaniaxx]

Absolutely loving these. I especially enjoyed the last third of the conversation. I hope to be much more mindful now, when applying "the mean" to the data at hand. Thank you for that.

[@colm_connaughton_lml]

Glad you are enjoying them. For this video, it was fun to talk about things that are "well known" but are sometimes forgotten or overlooked when encountered outside of the mathematics/statistics context.

[@II_superluminal_II]

omg KOLOMOGROV!!!!!!!!!!!!!!!!!!!!!!!! Complexity Theory was literally the only class that made me not drop out of CS...................... Kolomogrov complexity was something my professor helped create, which much much much easier to prove than pumping lemma. I mean who am I kidding, kolomogrov is the FATHER of complexity theory and probability and randomness and comprehensibility

[@RoryDavidWatts]

An excellent video, thank you Colm and Ole

[@UmarUmar-bg1fj]

Love these videos

[@jakeaustria5445]

Yey, another video in my recommendations.

[@prestonsumner7969]

Another excellent lesson. I would be interested in some examples of where the more esoteric means such as power mean and harmonic mean might appropriately apply.

[@colm_connaughton_lml]

It is often argued that harmonic mean is the "correct" mean to use when averaging quantities that are rates. Speed, for example, is the rate of distance travelled per unit time. If you travel the first half of a journey at speed u and the second half at speed v, then the total travel time would have been the same if you had travelled the entire journey at an "average speed" equal to the harmonic mean of u and v (not the arithmetic mean). Check out en.wikipedia.org/wiki/Harmonic_mean for lots of other examples.
The most widely used application of the power mean is probably the case 𝛾 = 2 which gives the root mean square. Every time you do a linear regression you are minimising a power mean of your errors - which doesn't seem so esoteric in that context!

[@lillix8826]

Beautiful stuff!

[@id3655]

Nice. Also, I hope you include more formalism in the ergodicity econ textbook.

[@olepeters6472]

Any specific aspect of the formalism you’re curious about? The textbook will start from first principles.

[@id3655]

@@olepeters6472 I will give an example... I have a good understanding of elementary mathematical analysis. In most books, the symbol "dt" does not refer to a separate mathematical object but are part of the traditional notation of calculus. They play no role in the definition of the derivative or integral. Regardless it's still easy to parse the meaning of most of basic differential equations. For me the problem is with the formulas of "stochastic differential equations" which I don't really understand... But even in ODE I was never taught to understand the meaning of dy = dx, it's just written like that so we can integrate, like syntactic sugar (yeah my course on DEs was absolute trash)... Anyway I recommend checking out Paul Wilmott's intro to quant. finance to see what NOT to do, see the first chapter intro to calc. See V.I. Arnold's ODE book digression on differential forms at the start of the book to see pedagogically correct approach... Why has Wilmot failed? Because the initiated will just have to skip those parts, and the uninitiated will surely not understand it at all and also end up skipping. I don't really know how to explain this better, hope you understand... Anyway maybe I'm not the target demo for the book but I think it's a useful thing to have in mind because authors are like "I want to make this book accessible" and end up making it difficult for us who do have a sketchy but bourbakist math background, and useless for the uninitiated just as well... Plenty of young (usually self-taught) ppl are like this and it makes it difficult to transition into applications... Ok... Well that's it, thanks for reading my rant... Best of luck with the book